3
Part of 2010 Portugal MO
Problems(2)
Portuguese food - Portuguese MO, Problem 3, 2010
Source:
8/28/2010
On each day, more than half of the inhabitants of Évora eats sericaia as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate sericaia as dessert.
combinatorics proposedcombinatorics
Primes and unit squares - Portuguese MO, Problem 6, 2010
Source:
8/28/2010
Consider a square , where is a prime number, which is divided by squares whose sides are parallel to the initial square's sides. Show that it is possible to select vertices such that there are no three collinear vertices.
analytic geometrymodular arithmeticnumber theory proposednumber theory